To determine which graph represents the function [tex]\( f(x) = \frac{1}{3} \tan(x) + 2 \)[/tex], we need to consider how the original tangent function [tex]\( y = \tan(x) \)[/tex] has been transformed.
1. Vertical Stretch/Compression:
[tex]\[
\frac{1}{3} \tan(x)
\][/tex]
- Here, the coefficient [tex]\(\frac{1}{3}\)[/tex] causes a vertical compression of the tangent function by a factor of [tex]\(\frac{1}{3}\)[/tex]. The peaks, valleys, and intercepts of the tangent function are reduced by a third of their typical distance from the x-axis.
2. Vertical Shift:
[tex]\[
+ 2
\][/tex]
- This transformation shifts the entire graph of [tex]\(\frac{1}{3} \tan(x)\)[/tex] up by 2 units.
Let's describe the transformation step-by-step:
- Start with the parent function [tex]\( y = \tan(x) \)[/tex], which has vertical asymptotes at [tex]\( x = \frac{\pi}{2} + k\pi \)[/tex] for any integer [tex]\( k \)[/tex].
- Apply the vertical compression: [tex]\( \frac{1}{3} \tan(x) \)[/tex] compresses all y-values of [tex]\( \tan(x) \)[/tex] to one third of their original value. The period remains [tex]\(\pi\)[/tex].
- Apply the vertical shift: by adding 2, every point on [tex]\( \frac{1}{3} \tan(x) \)[/tex] moves up 2 units.
So, the key characteristics of the graph [tex]\( f(x) \)[/tex]:
- The period remains [tex]\(\pi\)[/tex].
- The amplitude is scaled down by a factor of [tex]\(\frac{1}{3}\)[/tex].
- The asymptotes are still at [tex]\( x = \frac{\pi}{2} + k\pi \)[/tex], the same as the parent function [tex]\( \tan(x) \)[/tex].
- The whole graph is shifted vertically up by 2 units.
The corresponding graph should:
- Show the tangent function compressed vertically by a factor of 3.
- Be shifted upward by 2 units.
- Have vertical asymptotes at [tex]\( x = \frac{\pi}{2} + k\pi \)[/tex].
Unfortunately, without visual representations, I cannot select the graph directly. But by understanding the transformations, you can identify which one of the multiple choices demonstrates these characteristics:
1. Compressed tangent function (less steep slopes).
2. Upward shift by 2 units across the entire graph.
Look for these features in the available options to identify the correct graph representing [tex]\( f(x) = \frac{1}{3} \tan (x) + 2 \)[/tex].
